Theorem imaiun1 43913

Description: The image of an indexed union is the indexed union of the images. (Contributed by RP, 29-Jun-2020.)

Assertion

Ref	Expression
imaiun1	⊢ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶)

Distinct variable group:   𝑥,𝐶

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem imaiun1

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexcom4 3263	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
2		vex 3444	. . . . . 6 ⊢ 𝑦 ∈ V
3	2	elima3 6026	. . . . 5 ⊢ (𝑦 ∈ (𝐵 “ 𝐶) ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
4	3	rexbii 3083	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
5		eliun 4950	. . . . . . 7 ⊢ (⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑧, 𝑦⟩ ∈ 𝐵)
6	5	anbi2i 623	. . . . . 6 ⊢ ((𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ (𝑧 ∈ 𝐶 ∧ ∃𝑥 ∈ 𝐴 ⟨𝑧, 𝑦⟩ ∈ 𝐵))
7		r19.42v 3168	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ (𝑧 ∈ 𝐶 ∧ ∃𝑥 ∈ 𝐴 ⟨𝑧, 𝑦⟩ ∈ 𝐵))
8	6, 7	bitr4i 278	. . . . 5 ⊢ ((𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
9	8	exbii 1849	. . . 4 ⊢ (∃𝑧(𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
10	1, 4, 9	3bitr4ri 304	. . 3 ⊢ (∃𝑧(𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 “ 𝐶))
11	2	elima3 6026	. . 3 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
12		eliun 4950	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 “ 𝐶))
13	10, 11, 12	3bitr4i 303	. 2 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶))
14	13	eqriv 2733	1 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶)

Syntax hints:   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃wrex 3060  ⟨cop 4586  ∪ ciun 4946   “ cima 5627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-iun 4948  df-br 5099  df-opab 5161  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637

This theorem is referenced by:  trclimalb2  43988